A lemniscate is a figure-eight shaped curve that resembles the infinity symbol (∞). It is a type of mathematical curve defined by specific equations in polar or Cartesian coordinates. The most common form is the lemniscate of Bernoulli, which can be represented in polar coordinates as r^2 = a^2 \cos(2\theta) , where a is a constant. Lemniscates have interesting properties in geometry and are studied in various fields, including algebraic geometry and complex analysis. They can also be related to other mathematical concepts, such as ellipses and hyperbolas, showcasing the beauty of mathematical relationships and shapes.